Let $\varphi$ be the Euler totient function and $f$ be a nonnegative function such that
$$\sum_{n=1}^{\infty} \frac{f(n)\varphi(n)}{n}=\infty,$$
and that
$$\limsup_{N\to \infty} \frac{\sum_{n=1}^N f(n) \varphi(n)/n}{\sum_{n=1}^N f(n)} >0.$$
Let $g(n)=\min(f(n),\frac{n}{6\varphi(n)})$ (the digit $6$ is not crucial for this question), then do we have
$$\limsup_{N\to \infty} \frac{\sum_{n=1}^N g(n) \varphi(n)/n}{\sum_{n=1}^N g(n)} >0?$$
I know an estimate $\sum_{n=1}^N \frac{n}{\varphi(n)} = \frac{315\zeta(3)}{2\pi^4} N + O(x^{\epsilon}), \forall \epsilon>0$, which might be helpful here (this estimate is not optimal, though).
For those who are curious about the source of this question. This is taken from Harman's Metric Number Theory, page 37. What I asked is a gap in Harman's proof of Duffin-Schaeffer's theorem in 1941.